Suparno Bhattacharyya
Published: 2022
Total Pages: 0
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Simulating the dynamics of large-scale complex, spatio-temporal systems requires prohibitively expensive computational resources. Moreover, the high-dimensional dynamics of such systems often lacks physical interpretability. However, the intrinsic dimensionality of the dynamics often remains quite low, meaning that the dynamics remains embedded in a low-dimensional attractor or manifold in a high-dimensional state-space. Leveraging this phenomenon, in model order reduction, reduced order models (ROMs) with low-dimensional states are derived that can approximate the high-dimensional dynamics of large-scale systems with reasonable accuracy. In this thesis, we study the model reduction of structural systems subjected to impact and nonsmooth boundary conditions, using proper Orthogonal Decomposition (POD), a data-driven projection-based dimension reduction technique. The dynamics of structural systems is typically characterized by partial differential equations (PDEs), which are often impossible to solve analytically. A direct attempt to numerically solve these PDEs to obtain approximate solutions leads to extremely high-dimensional systems of ordinary differential equations (ODEs). The larger the dimensionality of the system of ODEs, the greater is the accuracy of the approximate solution. As a result, often, the dimensionality of a problem is artificially inflated to achieve a more accurate solution, even though the intrinsic dimensionality of the original system is much lower, making the problem computationally intractable. However, data from such high-dimensional systems often exhibit certain dominant patterns, which are representative of the underlying low-dimensional dynamics. POD identifies these low-dimensional embedded patterns based on the dominant correlations present in the data and determines a subspace that contains the data to a desired level of accuracy. This subspace is spanned by a set of basis functions known as proper orthogonal modes (POMs). Mathematically, the POMs are constructed such that along those the variance of the data is maximized. A certain number of POMs are chosen to form a reduced subspace onto which the high dimensional model of the system is projected, yielding a reduced order model that can parsimoniously describe the dynamics of the high-dimensional system. A major part of my research addresses the question of how best to determine the number of POMs to be selected, which is also the dimension of the ROM. In standard implementations of POD, this is decided such that a predefined percentage of the total data variance is captured. However, a fundamental problem with variance-based mode selection is that it is difficult, a priori, to determine the percentage of total variance that will lead to an accurate ROM. Furthermore, the needed percentage of variance can differ widely from one system to the next, or even from one steady-state solution to another. There are two main reasons for this. First, POD is essentially a projection-based technique that ensures optimal reduction (in a mean-square statistical sense) of high-dimensional data. However, such projection optimality does not ensure the accuracy of a ROM. This is because, second, the variance of a data set, or any portion of it in a reduced subspace, has no direct connection with the dynamics of the system generating it. In particular, dynamically important modes that have small variance can still play a crucial role in transporting energy in and out of the system. The neglect of such small-variance degrees of freedom can result in a ROM with behavior that significantly deviates from the true system dynamics. A specific aim of our work was to go beyond merely statistical characterizations to gain a physics-based understanding of why, in specific cases, a given dimension of the reduced subspace is required for an accurate ROM. We were particularly interested in dynamical systems that are subjected to nonsmooth loading conditions, such as impacts, or that have nonsmooth constitutive behavior, such as piecewise linear springs. Such features typically result in numerous modes being excited in the system dynamics. While performing model reduction of such systems, it is essential to include all dynamically important modes. We studied the model reduction of an Euler-Bernoulli beam that was subjected to periodic impacts, using a semi-analytical approach. It was observed that using the conventional variance-based mode selection criterion yielded ROMs with substantial inaccuracies for impulsive loading conditions, with a maximum of 5% relative displacement error and 50% relative velocity error. However, selecting the number of POMs required to achieve energy balance on the corresponding reduced subspace (the span of the selected POMs) gave ROMs with errors that were smaller by approximately three orders of magnitude. These ROMs properly reflect the energetics of the full system, resulting in simulations that accurately represent the system's true behavior. With variance-based mode selection, in principle one may always formulate ROMs with any desired accuracy simply by increasing the reduced subspace dimension by trial and error. However, such an approach does not provide any insight as to why this needs to be done in specific cases. The energy closure method provides this physical insight. We further studied the general application of this energy closure criterion using discrete data, with and without measurement noise, as typically gathered in experiments or numerical simulations. We used the same model of the periodically kicked Euler-Bernoulli beam and formulated ROMs by applying POD to the steady-state discrete displacement field obtained from numerical simulations of the beam. An alternative approach to quantifying the degree of energy closure was derived. In this approach, the convergence of energy input to or dissipated from the system was obtained as a function of the subspace dimension, and the dimension capturing a predefined percentage of either energy is selected as the ROM-dimension. This was in agreement with our prior idea of selecting the ROM dimension by ensuring a balance between the energy dissipation and input on the subspace since the steady-state dynamics guarantees that an accurate estimate of either quantity will automatically lead to a balance between the two. This new metric for quantifying the degree of energy closure was, however, found to be more robust to data-discretization error and measurement noise while also being easier to interpret. The data processing necessary for implementing the new metric was discussed in detail. We showed that ROMs from the simulated data using our approach formulated accurately captured the dynamics of the beam for different sets of parameter values. Finally, we implemented this new metric to estimate energy-closure for the model order reduction of an experimental system consisting of a magnetically kicked nonlinear flexible oscillator. This was a piecewise linear, globally nonlinear system, and exhibited a wide range of dynamical behaviors: periodic, quasi-periodic, and chaotic. Furthermore, the nonsmooth nature of the forcing and the boundary conditions excited a large number of modes in the system. For high-fidelity simulations, we approximated the dynamics of the oscillator using linear models with 25 degrees of freedom. By applying POD on the discrete displacement data obtained from the simulations and using the energy-closure criterion, we were able to formulate a single ROM, with only 6 degrees of freedom, which accurately captured the different dynamical steady states shown by the original system. More importantly, it was observed that ROM was able to preserve the bifurcation structure of the system. We have thus shown, how a physics-informed understanding of estimating ROM-dimension can lead to accurate reduced order models in linear and nonlinear structural vibration problems.
